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Suppose $X/\mathbb{Q}$ is a reasonable smooth projective variety with interesting etale cohomology. For example, we can say $X$ is an elliptic curve. To what extent does it make sense to study the etale cohomology of $X$ via Berkovich or adic spaces? (I'm asking as a total outsider to the field.) For example, does it make sense to try to learn something about $L(E,s)$ with these tools?

I read the introduction to Berkovich's 1993 IHES paper, but I didn't find any answers there. To what extent do Berkovich spaces allow you to say something about the Galois representation on the Tate module of $E$?

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The comparison theorem (analogous to Artin's over $\mathbf{C}$) shows that for coefficients in a constructible $\ell$-adic sheaf (and its analytification) one gets the same cohomology as in the algebro-geometric theory (for a separated scheme of finite type over a non-archimedean field), and likewise with proper supports. The relevance of the approach through non-archimedean geometry is when you try to compute cohomology of (or using) objects that are not analytifications of algebraic schemes. –  Marguax Sep 29 '13 at 1:09
    
@Marguax: Wonderful! Just to be super careful, when you say "the same cohomology" you are talking about as Galois modules, right. –  LMN Sep 29 '13 at 2:50
    
@LMN: Yes, but one can (and must) do so much better than that: the comparison ismorphisms are really at the level of R$f_{\ast}$'s and R$f_{!}$'s with constructible coefficients. The relativization is an essential feature of the proofs, as it is in the complex-analytic case (and Berkovich's proof, inspired by ideas of Deligne, yields a simpler proof for R$f_{\ast}$-comparison even in the complex-analytic case). –  Marguax Sep 29 '13 at 3:23
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@Marguax: How does it relate to the fact that the Berkovich space is contractible in the case of good reduction? –  Piotr Achinger Sep 29 '13 at 4:40
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@Piotr: Galois cohomology at the (henselian) residue fields of the points of the Berkovich spaces provides an "obstruction" to the equality of topological and etale cohomology (say for constant coefficients), made more precise by a spectral sequence, so purely topological features such as contractibility don't have the same consequences as over $\mathbf{C}$. This is all explained very clearly in Berkovich's IHES paper. I recommend that you read it. –  Marguax Sep 29 '13 at 6:13
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